In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang split...

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equatio...

One of the undesirable features in pistachio nuts (Pistacia vera L.) is the incidence of non-split nuts, which affects the price of the final product. In order to attain the kernel inside, the non-split nuts are mostly cracked mechanically. However, this practice may damage the kernel. Akbari variety of pistachio nut was selected to evaluate the influence of different non-mechanical treatments ...

In this lecture, we discuss splitting algorithms for convex minimization problems with objective given by the sum of two nonsmooth functions. We start with the fixed point property of such problems and derive a general scheme of splitting algorithm based on fixed point iteration. This covers Douglas-Rachford splitting and Peaceman-Rachford splitting algorithms. We also discuss the convergence r...

In this paper, we present the preconditioned mixed-type splitting iterative method for solving the linear systems, Ax = b, where A is a Z-matrix. And we give some comparison theorems to show that the convergence rate of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give a numerical example to illustrate our...